Hölder continuous dissipative solutions of ideal MHD with nonzero helicity

We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^α$ with $α=1/200$, for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for Hölder continuous solutions, is nonzero. The construction, which works both on the torus $\mathbb{T}^3$ and on $\mathbb{R}^3$ with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations.

💡 Research Summary

The paper establishes the existence of weak solutions to the three‑dimensional ideal magnetohydrodynamics (MHD) equations that are Hölder continuous with exponent α = 1/200, yet exhibit dissipation of the total kinetic‑magnetic energy and of the cross‑helicity (the so‑called Elsässer energies). Crucially, these solutions possess non‑zero magnetic helicity, which is forced to be conserved at every stage of the construction. This is the first instance of continuous weak solutions at a regularity level where one conserved quantity (magnetic helicity) is mathematically unavoidable, while another (energy or cross‑helicity) can be deliberately broken, and where the preservation of the former does not rely on any symmetry of the flow.

The authors begin by recalling the classical conservation laws of ideal MHD: the mean velocity and magnetic field (trivially conserved), the total energy E(t), the magnetic helicity H(t), and the cross‑helicity H×(t). Prior work shows that for Hölder regularity α > 1/3 the Elsässer energies are conserved, while magnetic helicity is already conserved for any α > 0. The challenge, therefore, is to construct solutions with α < 1/3 that still keep H constant but allow E and H× to vary. Existing convex‑integration schemes for MHD either produce symmetric solutions (hence zero magnetic helicity) or fail to control helicity at the desired regularity.

The key innovation is to embed the geometric “frozen‑in” property of magnetic field lines directly into the convex‑integration scheme. By Alfvén’s theorem, the magnetic field evolves as the push‑forward of its initial configuration under the volume‑preserving flow generated by the velocity. Lemma 2.1 formalizes this: if Xₜ denotes the flow map of v, then B(t) = Xₜ*B₀ for all t, and volume‑preserving diffeomorphisms automatically preserve magnetic helicity. Consequently, the authors design each iteration step around a carefully chosen volume‑preserving diffeomorphism ϕₜ that is close to the identity but carries high‑frequency oscillations.

The diffeomorphism ϕₜ is decomposed as ϕₜ = ϕ₀ₜ ∘ ϕ_cₜ. The leading part ϕ₀ₜ is an explicit, highly oscillatory map whose structure can be controlled; the correction ϕ_cₜ solves a fully nonlinear elliptic equation ensuring exact volume preservation. By selecting a “awkward” class of ϕ₀ₜ, the authors guarantee that ϕ_cₜ remains small, which is essential for subsequent estimates.

The velocity field v is then defined implicitly by the ODE ∂ₜϕₜ = v∘ϕₜ. This relationship yields a decomposition of v into an explicit component derived from ϕ₀ₜ and a remainder w_ϕ that encapsulates the intricate dependence on the correction diffeomorphism. The most delicate terms arise from the time derivative ∂ₜw_ϕ, which must be inverted through the operator ℛ (the inverse divergence). Standard Hölder spaces are insufficient to control these terms, so the authors introduce a Besov‑type scale where temporal and spatial derivatives are comparable. Within this framework they obtain precise bounds for w_ϕ, ∂ₜw_ϕ, and the associated Reynolds stress R.

At each iteration the Reynolds stress is split into three contributions: (i) a scaled-down version of the previous stress, (ii) cross terms generated by ϕ₀ₜ, and (iii) high‑frequency error terms stemming from w_ϕ and its time derivative. By a careful choice of frequency parameters and amplitude coefficients, the authors ensure that (ii) and (iii) are sufficiently small to be absorbed, while (i) decays geometrically. This yields a sequence of subsolutions converging to a genuine weak solution of the ideal MHD system.

To achieve compact spatial support (required for the statement on ℝ³), a final correction is applied. This involves composing the constructed subsolution with a localized, volume‑preserving compression map, thereby confining the support to a bounded domain without disturbing the previously established estimates.

The main theorem (Theorem 1.1) asserts that for any smooth, divergence‑free initial magnetic field (\bar B_0) with non‑zero helicity and any ε > 0, there exists a weak solution (v,B) ∈ C^{1/200}(ℝ³×